Expected Value and Variance, Stats 20 oct 7

X

A random variable: a numerical value determined by a random outcome

x

One possible value of X

P(X = x) or f(x)

Probability that X equals x: f(x) is the probability mass function (PMF)

Add over all possible values of X

Expected value, also called expectation or mean

Variance: a measure of spread around the mean

Standard deviation: the square root of variance

Constants: fixed numbers

A function of X, such as X²

Number of trials in a binomial distribution

Probability of success on each trial

Parameter of a Poisson distribution

Expected Value Formula

Multiply each value by its probability, then add

Expected square formula

Square each value, multiply by its original probability, then add

Variance definition formula

Subtract the mean from each value, square, multiply by probability, then add

Variance Shortcut formula

Expected square minus the square of the expected value

Standard deviation formula

Take the square root of variance

Expected function formula

Apply the function to each value, multiply by its original probability, then add

Expected value of a constant

Multiply by a constant

Add a constant

Add random variables

Subtract random variables

Combine scaling and addition

Variance / Constant C

Variance / Add constant C

Variance / Multiply by constant c

Variance / Add independent X,Y

Variance / Subtract independent X,Y

Variance / For independent X and Y

Distribution / Discrete uniform on 1,…,n

Distribution / Bernoulli(p)

Distribution / Binomial(n,p)

Distribution / Poisson

Boxes of Tickets

Check that the box matches the given probabilities before using its average. In your Q1, the box and the PMF had different proportions of 3s and 4s, so their means differed.

The variance of a single draw equals the box’s population variance, calculated using the total number of tickets \(n\) in the denominator. Sample variance uses n - 1